4 BALLS, SPHERES* DIAMETER 31

isometric space ll\ relating the distances of the images by/of the elements
of B which intervene in the theorem.

Let now E he a metric space, </ the distance on H, and /a bijection of H
onto a set 1C (where no previous distance need bo defined); we can then
define a distance cf on Ii' by the formula (3.3.1), and/is then an isoinetry
of H onto H'. The distance d* is said to have been transported from Ii to F/
by/

Example

(33.2) The e\tended real line R. The function /'defined in R by /(A*) ^
.v/(I f' |v|) in a bijection of R on the open interval I • [ • I, t IJ» the inverse

mapping y being defined by $( K} x }( I - |,v| ) for |AJ * I . I ,et J be the closed
interval [ 'I, f I J, and let R be the »et which h the union of R and of two
new clement* written H*» ami "• <*> (pointH at infinity); wo extend /to a
bijection of R onto J by putting /'(>«/>) -? H, f{ -•<* ), > • !, and write
ugain // for the* inverse mapping. As J to a metric for the Ufotance

|v "-- v|, we can apply the process described above to define ft as a metric
space, by putting i/U»r) -I/(v) /{rli, With thta cltHtunce (which, when
conHiiiercii for eleitirntH of Kt in different from the one defined in (3.2,1)), the
metric space R in culled the e\ tended mil line; we noli* that for A* > ()»

We can define an twder relation mi R by cidiiiifig v v; ,r iti be equivalent
to/(v) *% /'(r); it in readily vi*rifktil that for \» v in R thin i% eqaivitlcnt to
the order rchttum tin R» and itwst in addition we* have

- 'T- * i * IMI for every v*1 R; the mil iiijfiih«% are ;ilv» r»tilni the finite
elements of R All properties and deiitutunitt,, u*r« in (luipter II, which relate
ti» the order relation Wv (oxeluding everything which fiii% to do *ith algehntic
operations) Ciiii tititticdiatcly be Mtrdi$s|H>rted** to 1 by the mapping «/. A
nonempty A of R is ti/wm hounded l\n relation, and there*

fore sup A arul iiif A are defined, but may hi* I '/- i'»r -^ as well *i?» real
numl»erH, lite deftttitiott of sup w(x) and iiif n(,i| (for u of it set

» t- 4 i#4

A into R) is given in the same manner, aiul m particular, (2.3.5)»

(2.3.6), (2.3,7 ), ami luild without

4 BAM S,

Iii the theory of it is in use a

by of a

will be Ciiveit a I». il»and can very well consist only